Mathematical Economics Question, specific issues in the annex....

Mathematical Economics Question, specific issues in the annex.

Question 1 (12 points)
A. Find the total diferential in each o± the ±ollowing cases: (4 points)
y=(x+z)(x-2z) where x=2-7z
y= ax
2
+ bxz + cu, where x=eu+hv and z=gu where a,b,c w, h and g are constants.
Then ²nd the total derivative dy/dz in the ²rst equation, and the partial total
derivatives ∂y/∂u and ∂y/∂v in the second.
B. In each o± the ±ollowing cases, ²nd (dy/dx) using the implicit-±unction rule. (4 points)
F(y,x) = 3x
2
+ 2xy + 4y
3
= 0
F(y,x) = 12x
5
-2y= 0
F(y,x) = xy
2
e
y
C. For each o± the ±ollowing equations F(y, x)=0, is an implicit ±unction y=±(x) de²ned around
point x=1 and y=3? (4 points)
x
3
-2x
2
y +3xy
2
-22
=0 2x
2
+ 4xy -y
4
+ 67=0
Question 2 (8 points)
Consider the ±ollowing model o± the market in which the government taxes the production o±
the good.
Q
d
=
D(P)
Demand ±unction
Q
s
= S(P^,w
o
) Supply ±unction
Q
s
= Q
d
Equilibrium condition
P= P^ +t
o
The ±ollowing restrictions hold: D
P
< 0, S
P^
> 0, S
t
< 0, S
w
< 0
Here Q stands ±or quantity demanded/supplied, P ±or the price paid by consumers, P^ ±or the
price received by producers, t is the tax per unit on the good and w is the given wage o±
workers used in the production o± this good. Both wages and the tax are exogenous.
a. First eliminate P^ ±rom the model by substituting ±or P^ in the supply ±unction. Then use
the equilibrium condition to reduce the model to one equation, and write it in the ±orm:
F(endogenous variable, exogenous variables) =0. (2 points)
b. Veri±y whether the implicit ±unction theorem holds. (2 points)
c. Then determine the impact on equilibrium price P o± a change in (i) wages and (ii) taxes
using the implicit ±unction rule. (4 points)